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❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✐❛❧ ❙❝❤rö❞✐♥❣❡r ♦♣❡r❛t♦rs ▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❏♦✐♥t ✇✐t❤✿ ❆❧❡❦s❡② ❑♦st❡♥❦♦ ❛♥❞ ●❡r❛❧❞ ❚❡s❝❤❧ ❋❛❝✉❧t② ♦❢ ▼❛t❤❡♠❛t✐❝s ❯♥✐✈❡rs✐t② ♦❢ ❱✐❡♥♥❛ ❆✲✶✵✾✵ ❱✐❡♥♥❛ ❖❚■◆❉ ❱✐❡♥♥❛✱ ✶✾✳✶✷✳✷✵✶✻ ▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✶ ✴ ✶✹

✷ ■♥ ✱ ✷ ✶ ✶ ✷ ✷ ✷ ✐ ❉❡❝❛② ♣r♦♣❡rt✐❡s ♦❢ ❡ ❛s ❄ ✷ ✿ ❈♦♥s❡r✈❛t✐♦♥ ♦❢ ❝❤❛r❣❡✱ ✐ ■♥ ✷ ✳ ❡ ✷ ✵ ✵ ✿ ✵✱ ✵ ✫ ❉✐r✐❝❤❧❡t ❜✳❝✳✿ ✶ ✷ ✷ ✵ ✐ ❡ ✐ ❡ ✐ ❡ ✵ ✹ ✹ ✹ ✐ ✐ ✶ ❖❜✈✐♦✉s❧②✿ ✱ ✳ ❡ ✵ ✶ ✹ ◗✉❡st✐♦♥s✿ ❉♦❡s ❛ s✐♠✐❧❛r t✐♠❡ ❞❡❝❛② ❡①✐st ❢♦r ❢♦r✿ ✶ ✷ ❄ ✶ ✵❄ ✷ ❖t❤❡r ❜✳❝✳❄ ✸ Pr❡❧✐♠✐♥❛r✐❡s ▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✷ ✴ ✶✹

✐ ❉❡❝❛② ♣r♦♣❡rt✐❡s ♦❢ ❡ ❛s ❄ ✷ ✿ ❈♦♥s❡r✈❛t✐♦♥ ♦❢ ❝❤❛r❣❡✱ ✐ ■♥ ✷ ✳ ❡ ✷ ✵ ✵ ✿ ✵✱ ✵ ✫ ❉✐r✐❝❤❧❡t ❜✳❝✳✿ ✶ ✷ ✷ ✵ ✐ ❡ ✐ ❡ ✐ ❡ ✵ ✹ ✹ ✹ ✐ ✐ ✶ ❖❜✈✐♦✉s❧②✿ ✱ ✳ ❡ ✵ ✶ ✹ ◗✉❡st✐♦♥s✿ ❉♦❡s ❛ s✐♠✐❧❛r t✐♠❡ ❞❡❝❛② ❡①✐st ❢♦r ❢♦r✿ ✶ ✷ ❄ ✶ ✵❄ ✷ ❖t❤❡r ❜✳❝✳❄ ✸ Pr❡❧✐♠✐♥❛r✐❡s ■♥ L ✷ ( R + ) ✱ H = − d ✷ dx ✷ + l ( l + ✶ ) l ≥ − ✶ + V ( x ) = H l + V ( x ) , x ✷ ✷ ▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✷ ✴ ✶✹

✷ ✿ ❈♦♥s❡r✈❛t✐♦♥ ♦❢ ❝❤❛r❣❡✱ ✐ ■♥ ✷ ✳ ❡ ✷ ✵ ✵ ✿ ✵✱ ✵ ✫ ❉✐r✐❝❤❧❡t ❜✳❝✳✿ ✶ ✷ ✷ ✵ ✐ ❡ ✐ ❡ ✐ ❡ ✵ ✹ ✹ ✹ ✐ ✐ ✶ ❖❜✈✐♦✉s❧②✿ ✱ ✳ ❡ ✵ ✶ ✹ ◗✉❡st✐♦♥s✿ ❉♦❡s ❛ s✐♠✐❧❛r t✐♠❡ ❞❡❝❛② ❡①✐st ❢♦r ❢♦r✿ ✶ ✷ ❄ ✶ ✵❄ ✷ ❖t❤❡r ❜✳❝✳❄ ✸ Pr❡❧✐♠✐♥❛r✐❡s ■♥ L ✷ ( R + ) ✱ H = − d ✷ dx ✷ + l ( l + ✶ ) l ≥ − ✶ + V ( x ) = H l + V ( x ) , x ✷ ✷ ❉❡❝❛② ♣r♦♣❡rt✐❡s ♦❢ ❡ − ✐ Ht ❛s t → ∞ ❄ ▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✷ ✴ ✶✹

✵ ✵ ✿ ✵✱ ✵ ✫ ❉✐r✐❝❤❧❡t ❜✳❝✳✿ ✶ ✷ ✷ ✵ ✐ ❡ ✐ ❡ ✐ ❡ ✵ ✹ ✹ ✹ ✐ ✐ ✶ ❖❜✈✐♦✉s❧②✿ ✱ ✳ ❡ ✵ ✶ ✹ ◗✉❡st✐♦♥s✿ ❉♦❡s ❛ s✐♠✐❧❛r t✐♠❡ ❞❡❝❛② ❡①✐st ❢♦r ❢♦r✿ ✶ ✷ ❄ ✶ ✵❄ ✷ ❖t❤❡r ❜✳❝✳❄ ✸ Pr❡❧✐♠✐♥❛r✐❡s ■♥ L ✷ ( R + ) ✱ H = − d ✷ dx ✷ + l ( l + ✶ ) l ≥ − ✶ + V ( x ) = H l + V ( x ) , x ✷ ✷ ❉❡❝❛② ♣r♦♣❡rt✐❡s ♦❢ ❡ − ✐ Ht ❛s t → ∞ ❄ ■♥ L ✷ ✿ ❈♦♥s❡r✈❛t✐♦♥ ♦❢ ❝❤❛r❣❡✱ � � ❡ − ✐ tH f � ✷ = � f � ✷ ✳ � ▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✷ ✴ ✶✹

✐ ✶ ❖❜✈✐♦✉s❧②✿ ✱ ✳ ❡ ✵ ✶ ✹ ◗✉❡st✐♦♥s✿ ❉♦❡s ❛ s✐♠✐❧❛r t✐♠❡ ❞❡❝❛② ❡①✐st ❢♦r ❢♦r✿ ✶ ✷ ❄ ✶ ✵❄ ✷ ❖t❤❡r ❜✳❝✳❄ ✸ Pr❡❧✐♠✐♥❛r✐❡s ■♥ L ✷ ( R + ) ✱ H = − d ✷ dx ✷ + l ( l + ✶ ) l ≥ − ✶ + V ( x ) = H l + V ( x ) , x ✷ ✷ ❉❡❝❛② ♣r♦♣❡rt✐❡s ♦❢ ❡ − ✐ Ht ❛s t → ∞ ❄ ■♥ L ✷ ✿ ❈♦♥s❡r✈❛t✐♦♥ ♦❢ ❝❤❛r❣❡✱ � � ❡ − ✐ tH f � ✷ = � f � ✷ ✳ � H ✵ ✵ ✿ l = ✵✱ V ≡ ✵ ✫ ❉✐r✐❝❤❧❡t ❜✳❝✳✿ ✶ � ❡ ✐ ( x − y ) ✷ − ❡ ✐ ( x + y ) ✷ ❡ − ✐ tH ✵ √ � � ✵ f ( x ) = f ( y ) dy ✹ t ✹ t ✹ π ✐ t R + ▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✷ ✴ ✶✹

◗✉❡st✐♦♥s✿ ❉♦❡s ❛ s✐♠✐❧❛r t✐♠❡ ❞❡❝❛② ❡①✐st ❢♦r ❢♦r✿ ✶ ✷ ❄ ✶ ✵❄ ✷ ❖t❤❡r ❜✳❝✳❄ ✸ Pr❡❧✐♠✐♥❛r✐❡s ■♥ L ✷ ( R + ) ✱ H = − d ✷ dx ✷ + l ( l + ✶ ) l ≥ − ✶ + V ( x ) = H l + V ( x ) , x ✷ ✷ ❉❡❝❛② ♣r♦♣❡rt✐❡s ♦❢ ❡ − ✐ Ht ❛s t → ∞ ❄ ■♥ L ✷ ✿ ❈♦♥s❡r✈❛t✐♦♥ ♦❢ ❝❤❛r❣❡✱ � � ❡ − ✐ tH f � ✷ = � f � ✷ ✳ � H ✵ ✵ ✿ l = ✵✱ V ≡ ✵ ✫ ❉✐r✐❝❤❧❡t ❜✳❝✳✿ ✶ � ❡ ✐ ( x − y ) ✷ − ❡ ✐ ( x + y ) ✷ ❡ − ✐ tH ✵ √ � � ✵ f ( x ) = f ( y ) dy ✹ t ✹ t ✹ π ✐ t R + ❖❜✈✐♦✉s❧②✿ � ❡ − ✐ tH ✵ � L ✶ ( R ) → L ∞ ( R ) = ✶ ✹ π t ✱ t → ∞ ✳ √ ▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✷ ✴ ✶✹

✶ ✷ ❄ ✶ ✵❄ ✷ ❖t❤❡r ❜✳❝✳❄ ✸ Pr❡❧✐♠✐♥❛r✐❡s ■♥ L ✷ ( R + ) ✱ H = − d ✷ dx ✷ + l ( l + ✶ ) l ≥ − ✶ + V ( x ) = H l + V ( x ) , x ✷ ✷ ❉❡❝❛② ♣r♦♣❡rt✐❡s ♦❢ ❡ − ✐ Ht ❛s t → ∞ ❄ ■♥ L ✷ ✿ ❈♦♥s❡r✈❛t✐♦♥ ♦❢ ❝❤❛r❣❡✱ � ❡ − ✐ tH f � � ✷ = � f � ✷ ✳ � H ✵ ✵ ✿ l = ✵✱ V ≡ ✵ ✫ ❉✐r✐❝❤❧❡t ❜✳❝✳✿ ✶ � ❡ ✐ ( x − y ) ✷ − ❡ ✐ ( x + y ) ✷ ❡ − ✐ tH ✵ √ � � ✵ f ( x ) = f ( y ) dy ✹ t ✹ t ✹ π ✐ t R + ❖❜✈✐♦✉s❧②✿ � ❡ − ✐ tH ✵ � L ✶ ( R ) → L ∞ ( R ) = ✶ ✹ π t ✱ t → ∞ ✳ √ ◗✉❡st✐♦♥s✿ ❉♦❡s ❛ s✐♠✐❧❛r t✐♠❡ ❞❡❝❛② ❡①✐st ❢♦r H ❢♦r✿ ▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✷ ✴ ✶✹

✵❄ ✷ ❖t❤❡r ❜✳❝✳❄ ✸ Pr❡❧✐♠✐♥❛r✐❡s ■♥ L ✷ ( R + ) ✱ H = − d ✷ dx ✷ + l ( l + ✶ ) l ≥ − ✶ + V ( x ) = H l + V ( x ) , x ✷ ✷ ❉❡❝❛② ♣r♦♣❡rt✐❡s ♦❢ ❡ − ✐ Ht ❛s t → ∞ ❄ ■♥ L ✷ ✿ ❈♦♥s❡r✈❛t✐♦♥ ♦❢ ❝❤❛r❣❡✱ � � ❡ − ✐ tH f � ✷ = � f � ✷ ✳ � H ✵ ✵ ✿ l = ✵✱ V ≡ ✵ ✫ ❉✐r✐❝❤❧❡t ❜✳❝✳✿ ✶ � ❡ ✐ ( x − y ) ✷ − ❡ ✐ ( x + y ) ✷ ❡ − ✐ tH ✵ √ � � ✵ f ( x ) = f ( y ) dy ✹ t ✹ t ✹ π ✐ t R + ❖❜✈✐♦✉s❧②✿ � ❡ − ✐ tH ✵ � L ✶ ( R ) → L ∞ ( R ) = ✶ ✹ π t ✱ t → ∞ ✳ √ ◗✉❡st✐♦♥s✿ ❉♦❡s ❛ s✐♠✐❧❛r t✐♠❡ ❞❡❝❛② ❡①✐st ❢♦r H ❢♦r✿ l ≥ − ✶ ✷ ❄ ✶ ▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✷ ✴ ✶✹

❖t❤❡r ❜✳❝✳❄ ✸ Pr❡❧✐♠✐♥❛r✐❡s ■♥ L ✷ ( R + ) ✱ H = − d ✷ dx ✷ + l ( l + ✶ ) l ≥ − ✶ + V ( x ) = H l + V ( x ) , x ✷ ✷ ❉❡❝❛② ♣r♦♣❡rt✐❡s ♦❢ ❡ − ✐ Ht ❛s t → ∞ ❄ ■♥ L ✷ ✿ ❈♦♥s❡r✈❛t✐♦♥ ♦❢ ❝❤❛r❣❡✱ � � ❡ − ✐ tH f � ✷ = � f � ✷ ✳ � H ✵ ✵ ✿ l = ✵✱ V ≡ ✵ ✫ ❉✐r✐❝❤❧❡t ❜✳❝✳✿ ✶ � ❡ ✐ ( x − y ) ✷ − ❡ ✐ ( x + y ) ✷ ❡ − ✐ tH ✵ √ � � ✵ f ( x ) = f ( y ) dy ✹ t ✹ t ✹ π ✐ t R + ❖❜✈✐♦✉s❧②✿ � ❡ − ✐ tH ✵ � L ✶ ( R ) → L ∞ ( R ) = ✶ ✹ π t ✱ t → ∞ ✳ √ ◗✉❡st✐♦♥s✿ ❉♦❡s ❛ s✐♠✐❧❛r t✐♠❡ ❞❡❝❛② ❡①✐st ❢♦r H ❢♦r✿ l ≥ − ✶ ✷ ❄ ✶ V � = ✵❄ ✷ ▼❛r❦✉s ❍♦❧③❧❡✐t♥❡r ❉✐s♣❡rs✐✈❡ ❡st✐♠❛t❡s ❢♦r r❛❞✳ ❙❝❤r✳ ♦♣✳ ✷ ✴ ✶✹